Ideas of Michèle Friend, by Theme

[American, fl. 2007, Professor at George Washington University, Washington D.C.]

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2. Reason / D. Definition / 8. Impredicative Definition
An 'impredicative' definition seems circular, because it uses the term being defined
2. Reason / D. Definition / 10. Stipulative Definition
Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects
2. Reason / E. Argument / 5. Reductio ad Absurdum
Reductio ad absurdum proves an idea by showing that its denial produces contradiction
3. Truth / A. Truth Problems / 8. Subjective Truth
Anti-realist see truth as our servant, and epistemically contrained
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
In classical/realist logic the connectives are defined by truth-tables
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Double negation elimination is not valid in intuitionist logic
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic was developed for fictional or non-existent objects
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
A 'proper subset' of A contains only members of A, but not all of them
A 'powerset' is all the subsets of a set
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Set theory makes a minimum ontological claim, that the empty set exists
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Infinite sets correspond one-to-one with a subset
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Major set theories differ in their axioms, and also over the additional axioms of choice and infinity
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
Intuitionists read the universal quantifier as "we have a procedure for checking every..."
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
The powerset of all the cardinal numbers is required to be greater than itself
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
A number is 'irrational' if it cannot be represented as a fraction
The 'integers' are the positive and negative natural numbers, plus zero
The 'rational' numbers are those representable as fractions
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
The natural numbers are primitive, and the ordinals are up one level of abstraction
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
Cardinal numbers answer 'how many?', with the order being irrelevant
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
Raising omega to successive powers of omega reveal an infinity of infinities
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / k. Infinite divisibility
Between any two rational numbers there is an infinite number of rational numbers
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Is mathematics based on sets, types, categories, models or topology?
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Most mathematical theories can be translated into the language of set theory
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
The number 8 in isolation from the other numbers is of no interest
In structuralism the number 8 is not quite the same in different structures, only equivalent
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
Structuralist says maths concerns concepts about base objects, not base objects themselves
Structuralism focuses on relations, predicates and functions, with objects being inessential
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
'In re' structuralism says that the process of abstraction is pattern-spotting
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Mathematics should be treated as true whenever it is indispensable to our best physical theory
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism is unconstrained, so cannot indicate importance, or directions for research
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Constructivism rejects too much mathematics
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionists typically retain bivalence but reject the law of excluded middle
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Structuralists call a mathematical 'object' simply a 'place in a structure'
17. Mind and Body / E. Physicalism / 2. Reduction of Mind
Studying biology presumes the laws of chemistry, and it could never contradict them
18. Thought / D. Concepts / 1. Concepts / a. Concepts
Concepts can be presented extensionally (as objects) or intensionally (as a characterization)
Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability